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%I #17 Nov 10 2022 17:38:26
%S 1,2,4,10,24,56,136,328,784,1896,4576,11008,26592,64192,154752,373696,
%T 902144,2176640,5255424,12687488,30621952,73931392,178484736,
%U 430845952,1040176640,2511199232,6062209024,14635617280,35333443584,85300015104
%N Transform of 2^n by the aerated Catalan triangle A165408.
%C Hankel transform is A165410.
%H Vincenzo Librandi, <a href="/A165409/b165409.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: 1/(1-2*x-2*x^3*c(2*x^3)) = 2/(1-4*x+sqrt(1-8*x^3)) = (1-4*x-sqrt(1-8*x^3) )/(4*x*(1-2*x-x^2)), c(x) the g.f. of A000108.
%F G.f.: 1/(1-2*x-2*x^3/(1-2*x^3/(1-2*x^3/(1-2*x^3/(1-... (continued fraction).
%F a(n) = Sum_{k=0..n} if(n<=3k, 2^k*C((n+k)/2, k)*((3*k-n)/2 + 1)(1+(-1)^(n-k))/(2*(k+1)) = Sum_{k=0..n} 2^k * A165408(n,k).
%F a(n) = Sum_{k=0..n+1} Pell(n-k+1)*(0^k - 2^((k-2)/2)*A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3).
%F (n+1)*a(n) = 2(n+1)*a(n-1) + (n+1)*a(n-2) + 4*(2*n-7)*a(n-3) - 8(2*n-7)*a(n-4) - 4*(2*n-7)*a(n-5). - _R. J. Mathar_, Nov 17 2011
%F a(n) ~ (4+sqrt(2)) * (1+sqrt(2))^n / 8. - _Vaclav Kotesovec_, Feb 01 2014
%t CoefficientList[Series[2/(1-4*x+Sqrt[1-8*x^3]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *)
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2/(1-4*x+Sqrt(1-8*x^3)) )); // _G. C. Greubel_, Nov 10 2022
%o (SageMath)
%o def A165408(n,k): return 0 if (n>3*k) else (1+(-1)^(n-k))*(3*k-n+2)*binomial(int((n+k)/2), k)/(4*(k+1))
%o def A165409(n): return sum(2^k*A165408(n,k) for k in range(n+1))
%o [A165409(n) for n in range(41)] # _G. C. Greubel_, Nov 10 2022
%Y Cf. A000108, A000129, A165408, A165410.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 17 2009